The main purpose of this section has been to demonstrate the usefulness of
PDESystem,Amperslaw to problems in plasma physics, including effects due to the vacuum background.
However, the quantum relativistic generalization of Langmuir waves is
interesting in its own right, and the full dispersion function (REF ) will be
thoroughly investigated in a forthcoming paper. Here the vacuum polarization
contribution to (REF ) will be of much interest, and also the issue of pair-production, as induced by wave-particle interaction with the quantum vacuum.
As it turns out, a complete treatment of the quantum vacuum will require a renormalization, in order to remove the ultra-violet divergences [1]}, i.e. the high momentum divergences in the integrals chi1chi4. These divergences are of logarithmic type.
